There have been two methods for controlling magnetic levitation to move an object linearly in a horizontal plane:
(a) An electromagnet is mounted on a slider to levitate it magnetically above a ferromagnetic rail to move it horizontally along the rail. While the method provides a relatively easy way of controlling magnetic levitation because the point of action of a magnetic force on the slider is fixed constant with respect to the center of gravity of the slider, the method is disadvantageous in that the slider has to be moved pulling lead wires.
(b) A plurality of electromagnets are fixed and a ferromagnetic slider is levitated by magnetic fields produced by the electromagnets. This method allows the slider to be moved without pulling lead wires. However, since the position of the point of action of a magnetic force with respect to the center of gravity varies as the slider moves, magnetic levitation control is relatively difficult.
FIG. 7 shows a schematic diagram of a conventional horizontal linear slider device adapted to said method (b).
Axes of electromagnets 1, 2 are directed substantially vertically. Slider 3 is magnetically levitated vertically and moved horizontally in the first and second magnetic fields generated by the electromagnets 1, 2, respectively. Hereinafter, the vertical direction is referred to as the x direction, the horizontal direction in which the slider 3 moves as the y direction, and the direction perpendicular to both x and y directions as the z direction.
It is assumed that the first and second magnetic fields apply first and second magnetic forces f.sub.1, f.sub.2, respectively, to slider 3 at points of action, P.sub.1, P.sub.2, which are horizontally spaced apart by a distance a, that the center of gravity G of the slider 3 and the point of action P2 are horizontally spaced apart by a distance b, and that the gravity acting on the slider 3 is Mg. Then, the motion of the slider 3 can be divided into two components, i.e., the motion of the center of gravity caused by an external force F=f.sub.1 +f.sub.2 Mg and the rotational motion caused by a torque N=f.sub.1 (a-b)-f.sub.2 b about a rotational axis passing through the center of gravity and parallel to the z direction (hereinafter referred to as a "C.G. axis"). Thus, the process of controlling the magnetic levitation of the linear slider can be carried out by a center of gravity levitation control (hereinafter referred to as "C.G. levitation control") step to retain the center of gravity G at a given x position and by an inclination control step of controlling an angle of inclination .theta. of the slider 3 about the C.G. axis to hold the slider at a horizontal position (.theta.=0).
FIGS. 8 and 9 represent Laplace-transformed block diagrams to explain the fundamental concepts of said C.G. levitation control step and said inclination control step, respectively.
In the step shown in FIG. 8, the weight Mg/s of slider 3 is input as a disturbance into the control loop, and a magnetic force command p.sub.3 (s) is generated so that total magnetic force f.sub.1 (s)+f.sub.2 (s) balances the gravity Mg/s. A transfer function H.sub.4.sup.2 =K.sub.2 {1 +(T.sub.2 s).sup.-1 } is that of a proportional integral (PI) controller. Said magnetic force command p.sub.3 (s) generates a magnetic force f.sub.1 (s)+f.sub.2 (s) according to a transfer function H.sub.5.sup.2 =K.sub.F2. A transfer function H.sub.12.sup.2 =K.sub.D is the transfer function of an x.sub.G detector, and generates an x-position detection signal u.sub.G (s) from the x-position (x coordinate) of the center of gravity x.sub.G (s). The x-position detection signal u.sub.G (s) is compared with a preset value u.sub.G.sup.0 /S to generate a deviation signal .DELTA..sub.3 (s). These transfer functions constitute a closed control loop. The PI control unit outputs the magnetic force command p.sub.3 (s) so as to compensate for the deviation signal .DELTA..sub.3 (s). As can be calculated easily, a final value of the deviation signal ##EQU1## becomes 0, and the x-position of the center of gravity x.sub.G is controlled so that it is settled at the preset value u.sub.G.sup.0.
In the step shown in FIG. 9, it is assumed that said external force F exerted on slider 3 is substantially in balance (=0). Then, said torque N approximates f.sub.1 a-Mgb, i.e., the vector sum of torque caused by magnetic force f.sub.1 and gravity around an axis through point of action P2 parallel to the z direction (hereinafter, referred to as the P2 axis). Thus, control of inclination angle .theta. can be carried out so as to balance the torque f.sub.1 a with the torque Mgb. The torque Mgb(s) is input into the control loop as a disturbance. The resultant torque N=f.sub.1 (s)a-Mgb(s) generates an angle of inclination .theta.(s) according to a transfer function H.sub.11.sup.1 = 1/(Js.sup.2) (where J is the moment of inertia of the slider about the C.G. axis), and then generates an angle signal u.sub..theta. (s) corresponding to the angle of inclination .theta.(s) according to transfer function H.sub.12.sup.1 =K.sub..theta.. The difference .DELTA..sub..theta. (s) between the angle signal u.sub..theta.(s) and a preset angle value u.sub..theta..sup.0 (s) (=u.sub..theta..sup.0 /s) corresponding to the horizontal plane is input as an error signal to the PI controller (transfer function H.sub.4.sup.1), which then generates a torque command p.sub.1 (s)a (or a magnetic force command p.sub.1 (s)). The torque command p.sub.1 (s)a generates a torque about the P2 axis, f.sub.1 (s)a=K.sub.F1 p.sub.1 (s)a according to the transfer function H.sub.5.sup.1=K.sub.F1.
Said transfer function H.sub.12.sup.1 is the transfer function of the angle detector which detects the angle of an inclination of the slider 3. The transfer functions H.sub.4.sup.1, H.sub.5.sup.1 correspond respectively to the transfer functions H.sub.4.sup.2, H.sub.5.sup.2 shown in FIG. 8.
If said slider 3 moves at a constant velocity v in the y direction, then b=r+vt, and hence b(s)=(r/s)+(v/s.sub.2). As a result, the angle deviation signal .DELTA..sub..theta. (s) is expressed by the following equation, and its final value .DELTA..sub..theta.L does not become 0: ##EQU2##
As described above, since the final value of the deviation does not vanish in the inclination control step shown in FIG. 9, slider 3 is settled in an inclined state (in an angular offset state) and the offset depends on the velocity v of said slider 3, and in order to compensate for such an offset, it is necessary to control the inclination of the slider so that any deviation in the x-position of the center of gravity x.sub.G is not caused by the inclination control.